Optimal. Leaf size=157 \[ \frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.17, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47,
37} \begin {gather*} \frac {-B+i A}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(-B+2 i A) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}-\frac {(-B+2 i A) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{3/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}+\frac {((2 A+i B) c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {(2 A+i B) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.85, size = 103, normalized size = 0.66 \begin {gather*} \frac {i (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-3 A+(A+2 i B) \cos (2 (e+f x))+(-2 i A+B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{6 c^2 f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 151, normalized size = 0.96
method | result | size |
risch | \(-\frac {i A \,{\mathrm e}^{4 i \left (f x +e \right )}+B \,{\mathrm e}^{4 i \left (f x +e \right )}+6 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-3 i A +3 B}{12 c \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(110\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i A \left (\tan ^{4}\left (f x +e \right )\right )-i B \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{4}\left (f x +e \right )\right )+3 i A \left (\tan ^{2}\left (f x +e \right )\right )-2 A \left (\tan ^{3}\left (f x +e \right )\right )-i B \tan \left (f x +e \right )+i A -2 A \tan \left (f x +e \right )+B \right )}{3 f a \,c^{2} \left (i-\tan \left (f x +e \right )\right )^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(151\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i A \left (\tan ^{4}\left (f x +e \right )\right )-i B \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{4}\left (f x +e \right )\right )+3 i A \left (\tan ^{2}\left (f x +e \right )\right )-2 A \left (\tan ^{3}\left (f x +e \right )\right )-i B \tan \left (f x +e \right )+i A -2 A \tan \left (f x +e \right )+B \right )}{3 f a \,c^{2} \left (i-\tan \left (f x +e \right )\right )^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.68, size = 155, normalized size = 0.99 \begin {gather*} \frac {{\left ({\left (-i \, A - B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-7 i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 4 \, {\left (-i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 \, {\left (i \, A + B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, {\left (-i \, A - B\right )} e^{\left (i \, f x + i \, e\right )} + 3 i \, A - 3 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{12 \, a c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \tan {\left (e + f x \right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.73, size = 146, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (2\,A\,\sin \left (2\,e+2\,f\,x\right )+A\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-2\,B\,\cos \left (2\,e+2\,f\,x\right )-A\,3{}\mathrm {i}+B\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{6\,a\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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